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A287299 Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469). 1

%I

%S 0,0,0,0,0,1,0,0,0,1,2,0,0,0,2,1,0,1,2,2,0,0,2,2,1,1,3,0,1,1,4,3,0,2,

%T 2,2,0,3,4,3,1,2,6,3,1,0,5,4,2,2,4,3,0,2,3,5,0,1,3,4,3,2,4,3,3,4,5,4,

%U 0,2,5,5,0,4,6,2,1,1,7,3,1,2,7,4,2,4,5,5,1,3,6,5,1,3,6,6,3,4,4,4,2,4,7,6,3,1,4,4,0,4,6,5,2,2,7,5,2,1,7,8,4

%N Number of ways of writing n as a sum of a proper prime power (A246547) and a nonprime squarefree number (A000469).

%C Conjecture: a(n) > 0 for all n > 108.

%H Ilya Gutkovskiy, <a href="/A287299/a287299.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Squarefree.html">Squarefree</a>

%F G.f.: (Sum_{k>=1} x^A246547(k))*(Sum_{k>=1} x^A000469(k)).

%e a(26) = 3 because we have [25, 1], [22, 4] and [16, 10].

%t nmax = 120; CoefficientList[Series[(Sum[Boole[SquareFreeQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]) (Sum[Boole[PrimePowerQ[k] && ! PrimeQ[k]] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]

%o (PARI) x='x+O('x^120); concat([0, 0, 0, 0, 0], Vec(sum(k=1, 120, (issquarefree(k) && !isprime(k))*x^k) * sum(k=1, 120, (isprimepower(k) && !isprime(k))*x^k))) \\ _Indranil Ghosh_, May 23 2017

%Y Cf. A000469, A098983, A246547, A282192, A282290, A282318, A282947.

%K nonn

%O 0,11

%A _Ilya Gutkovskiy_, May 23 2017

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Last modified October 23 13:11 EDT 2018. Contains 316528 sequences. (Running on oeis4.)